{"id":241,"date":"2025-01-30T04:03:24","date_gmt":"2025-01-30T04:03:24","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/4-5-immediate-settlement-of-shallow-foundations\/"},"modified":"2026-03-16T13:59:31","modified_gmt":"2026-03-16T13:59:31","slug":"4-5-immediate-settlement-of-shallow-foundations","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/4-5-immediate-settlement-of-shallow-foundations\/","title":{"raw":"4.5 Immediate settlement of shallow foundations","rendered":"4.5 Immediate settlement of shallow foundations"},"content":{"raw":"Elastic settlement of soil subjected to uniform, flexible loadings can be simply calculated as the sum of the volumetric strains \u03b5<\/em>vol<\/sub> along the thickness of the compressible layer, \u0397<\/em>:\n\n(4.7)<\/strong> [latex]\\rho = \\int\\limits_0^{\\rm H} {{\\varepsilon _{vol}}dz}[\/latex]\n\nThe same basic concept applies for the estimation of settlement of shallow foundations. Nevertheless, we need to introduce some corrections in the above equation to account for the effect of the rigidity of the foundation, the embedment depth, and the shape of the footing, while appropriately considering drained or undrained conditions. These are discussed in the following paragraphs.\n\n
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4.5.1 Effect of footing rigidity: Immediate settlement and contact stresses<\/h2>\nDistortion of soil due to the application of an external load or pressure depends on the rigidity of the loaded area, and the type of soil (fine-grained or coarse-grained) i.e., whether immediately after application of a loading the soil will behave as drained or undrained. In case of rigid <\/em>footings (Figure 4.10<\/a>), settlement is of course uniform. The distribution of contact stresses below the footing is, nevertheless, highly non-uniform. Assuming that soil behaves as elastic material, contact stresses at the outer edge of the footing (x<\/em> = B<\/em>\/2, Figure 4.10<\/a>) would become infinite:\n\n(4.8)\u00a0<\/strong>[latex]\\Delta {\\sigma _z} = \\dfrac{{2{q_{ext}}}}{\\pi }\\dfrac{1}{{\\sqrt {1 - {{\\left( {\\dfrac{{2x}}{B}} \\right)}^2}} }}[\/latex]\n\nIn a real fine-grained<\/em> soil however, they are limited by its shear strength. The drained Young\u2019s modulus E\u2032 <\/em>of coarse-grained <\/em>soils depends on the degree of confinement, and this affects the distribution of stress underneath a loaded area. The fact that the drained Young\u2019s modulus E\u2032<\/em>\u00a0depends on the confining stress is mathematically expressed as:\n\n(4.9)<\/strong> [latex]E' = {E'_{ref}}{\\left( {\\dfrac{{p'}}{{{p_{ref}}}}} \\right)^m}[\/latex]\n\nwhere p\u2032 <\/em>is the mean effective stress p\u2032 <\/em>= (\u03c3<\/em>\u2032<\/em>1<\/sub>+<\/em>\u03c3<\/em>\u2032<\/em>2<\/sub>+<\/em>\u03c3<\/em>\u2032<\/em>3<\/sub>)\/3, \u0395\u2032<\/em>ref<\/sub><\/em> is the reference modulus value at a reference mean stress pref<\/sub><\/em>, while the exponent m<\/em> is considered usually m <\/em>= 0.5 for sands and m <\/em>= 1.0 for clays. In other words, the Young\u2019s modulus is non-linear function of mean stress in sands. Near the outer edges, where confinement (i.e., the mean stress) is less compared to the centre of the footing, the stresses are lower (Figure 4.10b<\/a>).\n\n[caption id=\"attachment_2430\" align=\"aligncenter\" width=\"600\"]\"The<\/a> Figure 4.10. Distribution of contact stresses below a rigid footing transferring uniform vertical pressure to (a) fine-grained and (b) coarse-grained soil.[\/caption]\n\nContact stress distribution under a flexible<\/em> loaded area is uniform, but the settlement profile varies as soil surface is free to deform, depending again on whether soil is fine-grained or coarse-grained (Figure 4.11<\/a>). The shape of the deformed surface in the case of saturated clay (undrained loading)<\/em> will be concave upward (Figure 4.11a<\/a>), as it is reasonable to assume a uniform value of Eu<\/sub><\/em> across the thickness of the compressible clay layer, as in Figure 4.10a<\/a>. On the other hand, the shape of the deformed surface in the case of a coarse-grained material (drained loading<\/em>) will be concave downward (Figure 4.11b<\/a>). That is because the Young\u2019s modulus of sands E\u2032<\/em> varies non-linearly with confining stress. As the latter are lower near the edge of the footing, compared to the stresses at its centre, the settlement developing near the edges will be higher than near the centre of the footing.\n\n[caption id=\"attachment_2431\" align=\"aligncenter\" width=\"600\"]\"The<\/a> Figure 4.11. Distribution of contact stresses below a flexible footing transferring uniform vertical pressure to (a) fine-grained and (b) coarse-grained soil.[\/caption]\n\n
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4.5.2 Effect of embedment depth<\/h2>\nThe pressure acting on the soil that will result in ground settlement is the additional<\/em> pressure \u0394qext <\/sub><\/em>imposed at the level of the foundation, on top of the pre-existing geostatic vertical effective stress at this level, \u03c3\u2032<\/em>z<\/sub><\/em>0<\/sub>. Settlement due to the stress \u03c3\u2032<\/em>z<\/sub><\/em>0<\/sub> has already taken place during soil deposition, before application of the loading.\n\n[caption id=\"attachment_4107\" align=\"aligncenter\" width=\"400\"]\"Schematic<\/a> Figure 4.12. Effect of embedment depth.[\/caption]\n\nIf the foundation level is at the ground surface, and no other loads exist (e.g., preloading) then \u03c3\u2032<\/em>z<\/sub><\/em>0<\/sub> = 0 and the additional pressure \u0394qext<\/sub><\/em> is equal to the pressure from the footing qext<\/sub><\/em> i.e., the applied force divided by the area of the footing Qext<\/sub><\/em>\/(Area). But in the common case where the topsoil is excavated for the construction of the foundation level at depth Df<\/sub><\/em> (Figure 4.12<\/a>) and soil is unloaded, the pre-existing vertical effective stress \u03c3\u2032<\/em>z<\/sub><\/em>0<\/sub> is equal to the geostatic stress at this level \u03c3\u2032<\/em>z<\/sub><\/em>0<\/sub> = \u03b3<\/em>Df<\/sub><\/em>,<\/sub> where \u03b3<\/em>\u00a0is the unit weight of the soil. In that case:\n\n(4.10)\u00a0<\/strong>[latex]\\Delta {q_{ext}} = {q_{ext}} - \\gamma {D_f}[\/latex]\n\nBesides, embedment of a foundation will further reduce the expected settlement as:\n